Question

The complex number z = 2 + i is the root of the polynomial z^4 – 6z^3 + 16z^2 - 22z + 15 = 0. Find the remaining roots c) Let z= √(3 - i) i) Plot z on an Argand diagram. ii) Let w = az where a > 0, a E R. Express w in polar form iii) Express w^8 in the form ka^n(x + i√y) where k,x,y E Z

Answer 1

since 2+i is a root, so is 2-i so (z^2-4z+5) is a factor.
z^4 – 6z^3 + 16z^2 - 22z + 15 = (z^2-4z+5)(z^2-2z+3)
use the quadratic formula to find the other two roots

√(3 - i) = 1.7553 - 0.2848i
√(3 - i) i = 0.2848 - 1.7553i
w = a * 1.7783 cis -0.1609
now finish it off. Not sure what the a has to do with anything. It's just a scalar multiplier.

Answer 2

Needs further calculation step by step.